The Goursat curves of order n are the curves having the symmetries of a regular polygon with n sides, i.e. for which the group of isometries that leave it invariant is that of this polygon, namely the dihedral group of order 2n.

A curve is therefore a Goursat curve of order n
iff it is invariant by a rotation of an n-th of a turn (and non-invariant by a rotation by a smaller angle) and it has an axis of symmetry, or iff it has exactly n axes of symmetry.

All the curves with polar equation
where f is -periodic (m and n are coprime) and even (or odd) are Goursat curves of order a multiple of n.Example opposite:
with n = 5, m = 3.

More generally, given a complex function f, with the same properties, the curve with complex parametrization is a Goursat curve of order a multiple of n (the previous case being the case where f is real).This case includes the epi-
and hypotrochoids
().The more general case gives the polytrochoids.Opposite, the tritrochoid obtained for .

Given f with the same properties, the curve with complex parametrization is, also, a Goursat curve of order a multiple of n.

The curves defined by an intrinsic equation:
with f even, L-periodic and such that is a rational number m/n that is not an integer (m and n are coprime) are Goursat curves of order a multiple of n. Furthermore, the integer m is the rotation index of the curve.Equivalent form:
with f even,-periodic with zero mean.Example opposite: n = 5, m = 3, .We get this way all the Goursat curves of order n and nonzero rotation index.

Classification of the spherical generic curves, Goursat curves of order n, having exactly n double points.

There are exactly 3 kinds, one of which is composed of curves with odd order n: